# Properties

 Label 1216.2.h.c Level $1216$ Weight $2$ Character orbit 1216.h Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1215,1216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.1215");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{2} q^{7} + 4 q^{9}+O(q^{10})$$ q + b1 * q^3 + b2 * q^7 + 4 * q^9 $$q + \beta_1 q^{3} + \beta_{2} q^{7} + 4 q^{9} + 2 \beta_{2} q^{11} - \beta_{3} q^{13} + 3 q^{17} + (2 \beta_{2} - \beta_1) q^{19} - \beta_{3} q^{21} - 3 \beta_{2} q^{23} - 5 q^{25} + \beta_1 q^{27} - \beta_{3} q^{29} + 2 \beta_1 q^{31} - 2 \beta_{3} q^{33} + 2 \beta_{3} q^{37} + 7 \beta_{2} q^{39} + 2 \beta_{3} q^{41} + 2 \beta_{2} q^{47} + 4 q^{49} + 3 \beta_1 q^{51} + \beta_{3} q^{53} + ( - 2 \beta_{3} - 7) q^{57} + 3 \beta_1 q^{59} + 8 q^{61} + 4 \beta_{2} q^{63} + \beta_1 q^{67} + 3 \beta_{3} q^{69} + 6 \beta_1 q^{71} - 11 q^{73} - 5 \beta_1 q^{75} - 6 q^{77} - 2 \beta_1 q^{79} - 5 q^{81} + 4 \beta_{2} q^{83} + 7 \beta_{2} q^{87} + 4 \beta_{3} q^{89} - 3 \beta_1 q^{91} + 14 q^{93} + 2 \beta_{3} q^{97} + 8 \beta_{2} q^{99}+O(q^{100})$$ q + b1 * q^3 + b2 * q^7 + 4 * q^9 + 2*b2 * q^11 - b3 * q^13 + 3 * q^17 + (2*b2 - b1) * q^19 - b3 * q^21 - 3*b2 * q^23 - 5 * q^25 + b1 * q^27 - b3 * q^29 + 2*b1 * q^31 - 2*b3 * q^33 + 2*b3 * q^37 + 7*b2 * q^39 + 2*b3 * q^41 + 2*b2 * q^47 + 4 * q^49 + 3*b1 * q^51 + b3 * q^53 + (-2*b3 - 7) * q^57 + 3*b1 * q^59 + 8 * q^61 + 4*b2 * q^63 + b1 * q^67 + 3*b3 * q^69 + 6*b1 * q^71 - 11 * q^73 - 5*b1 * q^75 - 6 * q^77 - 2*b1 * q^79 - 5 * q^81 + 4*b2 * q^83 + 7*b2 * q^87 + 4*b3 * q^89 - 3*b1 * q^91 + 14 * q^93 + 2*b3 * q^97 + 8*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{9}+O(q^{10})$$ 4 * q + 16 * q^9 $$4 q + 16 q^{9} + 12 q^{17} - 20 q^{25} + 16 q^{49} - 28 q^{57} + 32 q^{61} - 44 q^{73} - 24 q^{77} - 20 q^{81} + 56 q^{93}+O(q^{100})$$ 4 * q + 16 * q^9 + 12 * q^17 - 20 * q^25 + 16 * q^49 - 28 * q^57 + 32 * q^61 - 44 * q^73 - 24 * q^77 - 20 * q^81 + 56 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} + 7 ) / 7$$ (2*v^2 + 7) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 14\nu ) / 7$$ (v^3 + 14*v) / 7
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( 7\beta_{2} - 7 ) / 2$$ (7*b2 - 7) / 2 $$\nu^{3}$$ $$=$$ $$7\beta_1$$ 7*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1215.1
 1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i −1.32288 − 2.29129i
0 −2.64575 0 0 0 1.73205i 0 4.00000 0
1215.2 0 −2.64575 0 0 0 1.73205i 0 4.00000 0
1215.3 0 2.64575 0 0 0 1.73205i 0 4.00000 0
1215.4 0 2.64575 0 0 0 1.73205i 0 4.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.h.c 4
4.b odd 2 1 inner 1216.2.h.c 4
8.b even 2 1 304.2.h.b 4
8.d odd 2 1 304.2.h.b 4
19.b odd 2 1 inner 1216.2.h.c 4
24.f even 2 1 2736.2.k.k 4
24.h odd 2 1 2736.2.k.k 4
76.d even 2 1 inner 1216.2.h.c 4
152.b even 2 1 304.2.h.b 4
152.g odd 2 1 304.2.h.b 4
456.l odd 2 1 2736.2.k.k 4
456.p even 2 1 2736.2.k.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.b 4 8.b even 2 1
304.2.h.b 4 8.d odd 2 1
304.2.h.b 4 152.b even 2 1
304.2.h.b 4 152.g odd 2 1
1216.2.h.c 4 1.a even 1 1 trivial
1216.2.h.c 4 4.b odd 2 1 inner
1216.2.h.c 4 19.b odd 2 1 inner
1216.2.h.c 4 76.d even 2 1 inner
2736.2.k.k 4 24.f even 2 1
2736.2.k.k 4 24.h odd 2 1
2736.2.k.k 4 456.l odd 2 1
2736.2.k.k 4 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{2} - 7$$ T3^2 - 7 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 7)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 3)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$(T^{2} + 21)^{2}$$
$17$ $$(T - 3)^{4}$$
$19$ $$T^{4} + 10T^{2} + 361$$
$23$ $$(T^{2} + 27)^{2}$$
$29$ $$(T^{2} + 21)^{2}$$
$31$ $$(T^{2} - 28)^{2}$$
$37$ $$(T^{2} + 84)^{2}$$
$41$ $$(T^{2} + 84)^{2}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 12)^{2}$$
$53$ $$(T^{2} + 21)^{2}$$
$59$ $$(T^{2} - 63)^{2}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} - 7)^{2}$$
$71$ $$(T^{2} - 252)^{2}$$
$73$ $$(T + 11)^{4}$$
$79$ $$(T^{2} - 28)^{2}$$
$83$ $$(T^{2} + 48)^{2}$$
$89$ $$(T^{2} + 336)^{2}$$
$97$ $$(T^{2} + 84)^{2}$$